Yes, it is essentially this problem posed by Hume that Kant created Kantianism to answer.

This is the first essence of the categorical imperative, that our nature is not necessarily rational. We do not have all the true axioms and see all the conclusions contained in those axioms. We are capable of rationality and we experience this as a choice.

We are tempted, based on whatever our definition is in the moment, to do wrong, but we can overcome this temptation and do right.

Perfectly rational beings would not have such a temptation, or at least not the experience of it as a tension. The usual analogy is that mathematicians, when concluding the answer is 7, aren't tempted towards the 6, they are aware that they could just write 6 anyways but that would not be "giving into the 6" and would not change their understanding that the answer is 7; they have a choice, but it simply makes no sense to choose anything but the right answer. We are the student in this analogy who simply does not know if it 7 or 6, and is trying to figure it out, weighing other options such as giving up or just guessing or deciding the whole damn system stinks, and the clock is ticking.

The categorical imperative is first of all referencing this experience of right and wrong arising from our incomplete knowledge. We are perfectly happy doing something today, but tomorrow we come to understand something new and doubt the rightness of what we did yesterday. We are not perfectly rational, not only in the sense that we lack knowledge but also in the sense that we cannot simply immediately apply any new learning; we are creatures of habit which requires moral exertion to change; we are creatures of internal conflict which requires moral exertion to resolve; we are creatures with choices that require moral exertion to even understand in the beginning.

The content of the categorical imperative is then what is actually right actions. Perfectly rational beings would simply know and all agree wherever they are in existence. Our fate is to not know, to expand our understanding of coherent principles one at a time; it is slow, but there is no other right path to walk upon.

It is an imperative because it is something we should urgently do, it is categorical because it is not justified by reference to some other thing, it is the good in itself.

Yet, that's just a waste of time if anything because as we all agree the conclusion is contained within the axioms/premises. — TheMadFool

Yes, but it can still be amazingly hard to discover the path between conclusion (=theorem) and axioms/premises. Such path is the sequence of rewrite operations (the "proof") that bring back the conclusion back to its axioms/premises.

It has taken 350 years of trying by a large number of people to bring back Fermat's Last Theorem to the axioms/premises of number theory. The conclusion/theorem was known for centuries. The axioms/premises were known (at least implicitly) for centuries. The sequence of rewrite operations (the "proof") was impossible to find for centuries, until Andrew Wiles recently found it anyway. The corrected proof was published in 1995.

There are still outstanding conclusions/theorems that stubbornly resist both falsification and proof. A good example is the Riemann hypothesis. No counterexample has been found for 160 years but also no path/proof that brings it back to the axioms.

Hence, it is not because the axioms/premises and the theorem/conclusion are known that the path/proof is known too. It can be fiendishly hard (or even impossible) to discover that path/proof.

If you construct an abstract, Platonic world using axioms/premises expressed in first-order logic, then there will be perceivable patterns in that world that can neither be falsified nor be proven from its construction axioms. That is exactly what Gödel's first incompleteness theorem proves.

That is exactly what Gödel's first incompleteness theorem proves. — alcontali

This is what I forgot but Godel's theorems simply say that some truths are unprovable whether we have axioms or NOT. It's not relevant to my point that axioms contain within them all possible truths derivable therefrom.

Initially I had the impression that arguments, that includes all knowledge there was/is/will be, was an outwardly expanding set - explosion if you will - and I felt that knowledge was/is increasing.

This however is wrong. Knowledge is an inwardly contracting set - an implosion actually. Knowledge doesn't increase as every theorem provable is contained within the axioms of a given body of knowledge.

A good analogy is a microscope and a biological cell. The axioms is the cell itself and all theorems we derive from the axioms is like viewing the cell under the microscope- we see detail and different structures (new theorems) but it is, after all, the cell (axioms) itself.

If you construct an abstract, Platonic world using axioms/premises expressed in first-order logic, then there will be perceivable patterns in that world that can neither be falsified nor be proven from its construction axioms. That is exactly what Gödel's first incompleteness theorem proves. — alcontali

This is not what it proves.

This holds for only a finite set of axioms and axiom schemata, an important condition; as new unprovable truths or straightup undecidable propositions can always just then be added as a new axiom to then be provable.

You think Plato's content to be brooding up there with a finite amount of axioms? Have you no respect for your elders!

Which underlines the experience of knowledge that Kant arrives at: knowledge is not apart from moral exertion, knowledge is moral exertion. You will have to morally exert yourself to try to understand why you made such a trivial mistake ... or then I've made a trivial mistake and I will have to do the moral exerting.

NOTE: The rather dull picture of knowledge I've painted only applies to the deductive branch.

Knowledge derived with INDUCTION, as far as science is concerned, proceeds from some to all and science has led to, what people fancifully describe as, an explosion of knowledge.

Although the utility of induction lies in deduction being applicable to knowledge thus gained we can at least claim an increase in knowledge.

This holds for only a finite set of axioms and axiom schemata, an important condition; as new unprovable truths or straightup undecidable propositions can always just then be added as a new axiom to then be provable. — boethius

If a pattern emerges inside a world constructed by a particular set of axioms, and this pattern stubbornly resists falsification, i.e. counterexamples are not forthcoming, but also resists being proven from the axioms, then adding this pattern to the axioms would be redundant. The pattern already emerges from the existing axioms without explicitly adding the pattern as an axiom. So, what do you gain by adding it?

For example, the Riemann hypothesis has resisted falsification for 160 years, i.e. counterexamples are not forthcoming. At the same time, this pattern has also resisted being proven from the axioms of number theory. Imagine that you add the Riemann hypothesis to the axioms and suddenly someone unexpectedly finds a counterexample. Now your system is inconsistent, because the axiom says that the counterexample does not exist, while someone just produced such counterexample.

Therefore, I really do not see what you would gain by adding the Riemann hypothesis to the axioms of number theory.

Well, this got pretty long, because there's lot's of important distinctions that we usually don't draw attention to.

To summarize, there are (at least 3) kinds of unproven statements.

The first, is as you say, statements that resist refutation. Both their truth value and undecideability value are unknown to us. Providing a refutation resolves both questions, "it is a decidable statement and it is not-true".

Continuing to not have a refutation has more possibilities than just "we haven't proven it true or false yet".

The statement could then be proven to be undecideable: which means we prove it neither follows from nor does it contradict our existing axioms in the system under consideration. It is here that we are safe to extend the system by just adding it as an axiom.

We might do so for fun, or we might do so because we have reasons, outside the system, to believe the statement really is true. Incompleteness demonstrates just this; we can't prove it ... but we really do think it's true!

Incompleteness informs us that we can never have a set of rules that prove "all true facts" about numbers, even if we keep finding undecideable statements and each time have reasons to believe it's true and so allowing more true proofs about numbers (to extend the "true facts" we know). But only because we are finite.

If you're talking about the platonic world with all true statements, mathematical or just statements in general, then incompleteness isn't a problem.

The pattern already emerges from the existing axioms without explicitly adding the pattern as an axiom. So, what do you gain by adding it? — alcontali

I'm not sure I would phrase things the way you do (as undecidable statements are not necessarily existing patterns), but adding a "true unprovable" statement as an axiom allows the system to be extended by then proving more things with the new axiom.

For instance, the axiom of choice is not provable from the other ZF axioms (it neither contradicts ZF nor is implied by ZF nor is it a pattern that emerges from statements following from ZF), so we can add it and extend the system. This is probably the most famous example, but we can also just take other axioms away from ZF and have the same situation and then add them back in to extend the system, to demonstrate the process (taking an axiom out, doesn't make the things depending on that axiom suddenly false, they become suddenly undecidable); something controversial like the axiom of choice (the controversy being not that we can extend ZF but whether we need to for the practical problems of engineers) isn't required to see this "adding axiom" process happening, that's the only way to get a set of axioms: staring with one axiom and then adding the next.

This "extending the axioms" also occurs whenever problems with actual values are worked out with defined relations or values. Saying X really is 5 extends our axiomatic system. This may seem trivial and irrelevant, but I'll get back to it at the end (we may have reason to really believe X is 5; i.e. that X represents the number of people in the room, and the number of people is really 5: it is a true statement we are adding to the system, as an axiom, that we could not prove with our previous axioms; if we are using this statement in our formal system it is no different than the other axioms, it is only us, outside the system, that knows it is a different kind than the others and stated to be true for different reasons; so we don't call them axioms, but they are formally they are the same thing and we demarcate that with "if"; i.e. "if x = to 5" then we are going to treat that as an axiomatic statement ... for now).

Therefore, I really do not see what you would gain by adding the Riemann hypothesis to the axioms of number theory. — alcontali

The reason we don't axiomatize the Riemann hypothesis is because, unlike the above two examples, we really don't know if there are no counter examples; if we add the hypothesis as an axiom and a counter example is found then now we have a contradiction and all statements (all statements!) and their negations can be proved; we want to avoid that. When the Riemann hypothesis is "assumed" it's being used as an axiom same as every other, but again, we outside the system know it's not like the others, and if a problem arises (a contradiction appears) we won't be pointing the finger at axioms at random, we'll know who needs to go.

For the Riemann hypothesis to be added like the axiom of choice, it would need to be proven that it's undecidable; i.e. independent of ZFC. Then we could add and make ZFCR or it's negation or do neither no problems, and it's a practical question whether it's useful to add it and it's a philosophical debate whether it is really "true" (formally speaking, in the same way we can ask if ZFC axioms are "really true"; though from outside the system we can understand ZF is different than the C which is different than R) -- useful to note here most mathematicians no longer debate whether axioms are true or not; you do what you want and you see what happens, if you want to do something useful pick useful axioms ... of course, that's not how mathematics is taught. (Another good example is imaginary numbers, we need to add i squared equals 1 as an axiom -- and for that matter analytic continuation from which the Riemann hypothesis arises is also itself an extension, adding more axioms because we feel like it; why it's such a focus is that a bunch of other stuff about prime numbers and number theory become true making a deep and unexpected connection; this is what's irksome, having to jump through all these hoops to be really, really close to proving things we have not the slightest clue how to do otherwise).

Anyways, once we understand all that we can extend our mathematical axioms by making new axioms from undecideable statements, and we can imagine axiomatizing all knowledge through this process as the OP suggests. For instance, "at what time you'll wake up tomorrow" is not provable from our current set of axioms, but once it happens we can add it as an axiom to a giant formal system we're continuously extending as we think of new statement we prove are undecideable but think are true anyways as well as experience new things that get dropped into the system and true because they happened (they didn't have to -- i.e. we couldn't prove it from previous axioms -- but it did happen and so becomes a true statement we can use as an axiom). A "perfectly rational" being with "all the axioms" would indeed see all the conclusions in the axioms and experience knowledge in this way (an omniscient being would have no subtleties about what's true and false; a perfectly rational but not-omniscient would just have perfectly accurate probabilities that follow from any uncertainty in their axioms; i.e. we can interpret "all the axioms" in an omniscient way or in a way of perfectly setting up all the experience the being has as axioms and making perfect inferences).

Of course, we're far from being able to do anything remotely close to this.

This is why I describe Kant, not only because of the historical parallel, but because the fact we are so far away from experiencing "real knowledge" in the way the OP suggests (that I agree, "real perfect knowledge" works like that; all the conclusions are understood simultaneous to the axioms) "our actual experience" of knowledge is the moral effort required to understand a tiny, small, miniscule part of the "platonic" world of all truths. Because we can get it wrong along the way gives rise to moral tension. The Kantian philosophy is that there is a path -- there are true axioms that can be discovered and we can through effort conform our behaviour to those axioms approaching, in steps no matter how small, the world view of perfectly rational beings for whom it is just obvious and there is no tension -- which is opposed to nihilism of no true axioms existing (at least morally), skepticism of not true axioms being knowable but they maybe there, relativism of one form or another where true axioms depend on oneself (in a circuitous and unresolvable way ... unless it is already resolved), divine consequentialism (true axioms are decided by God and only true by being told to do it or suffer the consequences to disobey), utilitarianism, scientism, emotivism (where things aren't resolvable at all).

Interesting - this is more or less what I had in mind while advancing solutions for the original sin and good/evil: apply them as (if) an axiom in order to see if they map out onto creation such that it describes what we see.

Unfortunately these threads were removed without explanation, though it would have been interesting to have discussed.

I find there are axioms that, when applied/understood, render a way of seeing the creation such that one is no longer polarized (ie. for/against anything or anyone in particular) and they are able to see things just the way they are without distortion. I'd like to discuss these but, again, my posts are being removed so am awaiting clarification as to why.

If it's true that all we need to know is axioms, then the question becomes, which axioms? We know that in math, at least, axioms are insufficient to characterize mathematical truth. Modern set theory is about the search for intuitively appealing axioms that resolve the questions we care about (Continuum hypothesis for example). In the end the choice of axioms is social and political, not just a matter of rationality. Do you want more restrictive axioms or more permissive ones? There's no right or wrong answer.

To be frank all I understand about Godel's incompleteness theorems is that there are some unprovable truths in any axiomatic system. In the discussion the two of you were having a word, "undecidable" was used. Kindly clarify the difference, if any, between "unprovable" and "undecidable".

If I were to make an educated guess, "unprovable" and "undecidable" mean the same thing.

If I were to make an educated guess, "unprovable" and "undecidable" mean the same thing. — TheMadFool

Undecidable is actually stronger. It means not only unprovable, but also that the negation is unprovable.

I've read some logic and the video gets along well with what standard textbooks say:

1. A sound deductive argument is one in which the conclusion follows necessarily from the premises. — TheMadFool

No, that's validity. Soundness is where the argument is valid and the premises and conclusion are true.

Undecidable is actually stronger. It means not only unprovable, but also that the negation is unprovable. — alcontali

:ok:

Undecidable = neither provable nor disprovable???

No, that's validity. Soundness is where the argument is valid and the premises and conclusion are true. — S

I corrected my post. Thanks.

Undecidable = neither provable nor disprovable??? — TheMadFool

Yes. ;-)

Can you have a look at Godel's Incompleteness Theorems vs Justified True Belief

@TheMadFool, can you also please look at the thread you invoked just here?

Can you have a look at Godel's Incompleteness Theorems vs Justified True Belief — TheMadFool

First of all, there is an enormous problem with knowledge as a Justified (true) Belief. Mathematics is not correspondence-theory "true". In fact, mathematics has not necessarily anything to do with the real, physical world. Mathematics does not "correspond" to the real, physical world.

Mathematics is about patterns in abstract, Platonic worlds constructed from their basic axioms. For some of these patterns it is possible to demonstrate that they necessarily follow from their Platonic world's construction logic. Such demonstration is a "proof".

In more practical terms, the term "provable" means that there exists a sequence of rewrite transformations that will connect a theorem to the construction axioms of its world. It has nothing to do with correspondence-theory "true", which is a real-world, physical concept. Hence, mathematics are justified beliefs, but not justified true beliefs.

Furthermore, logically "true" does not mean correspondence-theory "true".

Using a clever hack, Gödel manages to create a theorem that is algebraically "true" in the abstract, Platonic world of number theory, but which by simply asserting its own unprovability, is not provable from the construction logic of that world. Hence, true but not provable.

The Gödel statement says that it is not provable. So, it isn't. What is says, is therefore true.

Concerning the real, physical world, we cannot prove anything at all, because we do not have access to its unknown construction logic. Therefore, every correspondence-theory "true" statement is always not provable. Therefore, in the real, physical world, "true" always implies "not provable".

Stephen Hawking argued that Gödel's incompleteness theorem has an effect on science -- which seeks to be correspondence-theory "true" -- and will lead to undecidable statements in science too. It would be nice if someone came up with a witness statement for that, though.

Hence, mathematics are justified beliefs, but not justified true beliefs. — alcontali

Using a clever hack, Gödel manages to create a theorem that is algebraically "true" in the abstract, Platonic world of number theory, but which by simply asserting its own unprovability, is not provable from the construction logic of that world. Hence, true but not provable. — alcontali

Indeed. I sensed that there wasn't the required level of correspondence between Godel's incompleteness theorems (GIT) and the justified true belief (JTB) of philosophy. The shared characteristic between the two I was hoping to emphasize was the need for proof to establish truth. As you can see I'm under the impression that the only way to claim truth in mathematics is by proof. Hence my puzzlement when "[u[true[/u], but unprovable" cropped up in GIT.

Can you tell me more about

algebraically "true" — alcontali

?

My logic:

If a proposition P is true then necessarily that a proof must exist for P being true.

Mx = x is a mathematical proposition
Px = x has a proof
t = mathematical proposition that Godel says is true BUT not provable.

Mt = t is a mathematical theorem
~Pt = not the case that t has a proof
1. Ax(Mx > Px)....premise
2. Mt.......premise
3. ~Pt......premise
4. Mt > Pt.......1 UI
5. Pt...........2, 4 MP
6. Pt & ~Pt....contradiction

As you can see I have to reject one or all of the three propositions 1, 2, or 3.

If I were to make an educated guess, "unprovable" and "undecidable" mean the same thing. — TheMadFool

I think this question got answered, but I was careful to use terms of "unproven statements" which is not the same as "unprovable statements.

Why we don't use "unprovable" as a synonym to undecideable is because it makes sense to say "this false statement is unprovable", but of course to know it's false means the negation is proven. Which is why "Undecidable is actually stronger. It means not only unprovable, but also that the negation is unprovable" as @alcontali mentions.

:up:

If a proposition P is true then necessarily that a proof must exist for P being true. — TheMadFool

This sounds like an endorsement of constructivist logic, such as intuitionistic logic, which requires a positive proof in order to affirm any proposition and accordingly denies the Law of Excluded Middle (LEM, either A or not-A must be true). By contrast, classical logic affirms LEM, which is why it allows double negation elimination (not not-A implies A) and proof by contradiction (reductio ad absurdum).

This sounds like an endorsement of constructivist logic, such as intuitionistic logic, which requires a positive proof in order to affirm any proposition and accordingly denies the Law of Excluded Middle (LEM, either A or not-A must be true). By contrast, classical logic affirms LEM, which is why it allows double negation elimination (not not-A implies A) and proof by contradiction (reductio ad absurdum). — aletheist

:up: I wasn't excluding reductio ad absurdum as a method of proof. I'm questioning what appears paradoxical to me in Godel's Incompleteness Theorems viz: that there are theorems that are true but unprovable. How is that possible?

The works of Kurt Gödel, Alan Turing, and others shook this assumption, with the development of statements that are true but cannot be proven within the system. — Wikipedia

A proof is sufficient evidence or a sufficient argument for the truth of a proposition. — Wikpedia

On one hand mathematicians are of the view that truth requires proof and on the other they're claiming, through Godel, that some truths are unprovable. Isn't that a contradiction?

On one hand mathematicians are of the view that truth requires proof and on the other they're claiming, through Godel, that some truths are unprovable. Isn't that a contradiction? — TheMadFool

No, the two Wikipedia quotes are not contradictory. The second one only affirms that a proof is sufficient for the truth of a proposition; it does not state that a proof is necessary for the truth of a proposition.

No, the two Wikipedia quotes are not contradictory. The second one only affirms that a proof is sufficient for the truth of a proposition; it does not state that a proof is necessary for the truth of a proposition. — aletheist

Ok. I think wikipedia may have some errors. That's a big accusation but consider the following

1. Proof implies truth

You agree on that (proof as a sufficient condition for truth)

What was the sufficient proof for the Godel statement "this statement is true but unprovable"?

Also, it seems that you're denying my claim: truth of a proposition necessarily requires proof.

2. If proposition true then proof exists. You're denying this.

That means it's possible for a proposition to be true and without proof.

Can you name one such truth?

How do you know it's true?

What was the sufficient proof for the Godel statement "this statement is true but unprovable"? — TheMadFool

That would be Gödel's proof of his incompleteness theorem, and the correct term is not "unprovable" but undecidable.

Also, it seems that you're denying my claim: truth of a proposition necessarily requires proof. — TheMadFool

In this context, I am not so much denying it as pointing out that it does not apply to all systems of formal logic. Informally, the claim is obviously false, since lots of propositions are true without ever having a formal proof. It seems like you may be confusing truth with justification.

That means it's possible for a proposition to be true and without proof. Can you name one such truth? — TheMadFool

Can you provide a proof that the truth of a proposition necessarily requires proof? If not, why do you claim that?

On one hand mathematicians are of the view that truth requires proof and on the other they're claiming, through Godel, that some truths are unprovable. Isn't that a contradiction? — TheMadFool

Can there be proof for the proposition X="There is no proof for X in theory T" in theory T? If there is proof for X in theory T, then theory T is inconsistent. If there is no proof for X in theory T, then X is true. That makes proposition X "true but not provable".

The fact that proposition X is even true, is actually not even the main problem.

The main problem is: Do there exist yes/no questions in T for which T cannot prove a yes or a no answer? The existence of such questions would be the "incompleteness" of T.

Answer: If the language of T does not allow for expressing proposition X, then the problem does not even apply. If it does, however, then there is at least one yes/no question in the language of T, namely proposition X, for which T cannot prove/decide the answer, irrespective of the axioms in T. Furthermore, the problem cannot be fixed by re-engineering the axioms of T either.

Can you provide a proof that the truth of a proposition necessarily requires proof? If not, why do you claim that? — aletheist

Your question, asking for proof without which you won't be satisfied, is proof that claims necessarily require proof.

No, my question demonstrates that a proof is not necessary for a proposition to be true. It is self-refuting to claim otherwise, unless you can provide a proof of that claim. Again, a proof is a form of justification, while truth is a property of propositions themselves.

Thank you. I think we're both right.

The burden of proof (Latin: onus probandi, shortened from Onus probandi incumbit ei qui dicit, non ei qui negat) is the obligation on a party in a dispute to provide sufficient warrant for their position. — Wikipedia

A given proof is a sufficient condition to establish truth AND it's necessary too. I'm concerned about the latter aspect viz. necessity. You're right too.

:up: :ok:

A given proof is a sufficient condition to establish truth AND it's necessary too. I'm concerned about the latter aspect viz. necessity. — TheMadFool

No, a proof is sufficient but NOT necessary. A true proposition is true regardless of whether humans ever construct a proof for it. One more time: a proof pertains to justification, not truth.